Algebro-geometric solutions for the Hunter–Saxton hierarchy

Abstract

This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the Hunter-Saxton (HS) hierarchy through studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the HS hierarchy, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire HS hierarchy.

Symmetry-Preserving Difference Models of Some High-Order Nonlinear Integrable Equations

“…Hunter-Saxton equation is firstly derived by Hunter and Saxton to describe the weakly nonlinear hyperbolic waves governed by variational principles [33]. In 2014, Hou et al [34] derived the Hunter-Saxton hierarchy, in which the equation is given as Furthermore, the Hunter-Saxton can be seen as the high frequency limit of the Camassa-Holm equation [35][36][37] (20)…”

Algebro-Geometric Solutions and Their Reductions for the Fokas-Lenells Hierarchy

“…This systematic approach, proposed by Gesztesy and Holden to construct algebro-geometric solutions for integrable equations, has been extended to the whole (1+1) dimensional integrable hierarchy, such as the AKNS hierarchy, the CH hierarchy etc. Recently, we investigated algebro-geometric solutions for the the Degasperis-Procesi hierarchy and Hunter-Saxton hierarchy [9,10] using this method.…”

Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

“…It also describes the geodesic flow on the homogeneous space related to the Virasoro group [33,34,35]. Recently, algebro-geometric solutions for the HS hierarchy was investigated in [28]. Moreover, peakon solutions, global weak solutions and the Cauchy problem of the system (1.1) were discussed in [9,23,36,44].…”